Bounding $|\cos (2n+1)\theta |$

Question

I am trying to find a bound for $|\cos (2n+1)\theta |$ by something of the form $c(n)|\cos \theta|$. I am suspecting that we can bound it by $(2n+1)|\cos \theta|$ but I'm not sure how to show that this is true.

Answer 1

Use $2\cos t=e^{it}+e^{-it}$. Then $$\frac{\cos(2n+1)t}{\cos t}=\frac{e^{(2n+1)it}+e^{-(2n+1)it}}{e^{it}+e^{-it}}=e^{2nit}-e^{2(n-1)t}+e^{2(n-2)it}-\cdots+e^{-2nit}.$$ As each of these exponentials is bounded by $1$ is absolute value, you get $$|\cos(2n+1)t|\le(2n+1)|\cos t|.$$ This is the best possible; take $t$ close to $\pi/2$, then $e^{-2it}$ is close to $-1$ etc.

Answer 2

Well since $\cos(0)=1$ I'm not sure you can do better than $c(n)=1$. (although you can bound $\sin n\theta$ or complex $\cos n\theta$ non-trivially)